<h2>Problem 127</h2>
<div style="color:#666;font-size:80%;">01 September 2006</div><br />
<div class="problem_content">
<p>The radical of <i>n</i>, rad(<i>n</i>), is the product of distinct prime factors of <i>n</i>. For example, 504 = 2<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> 3<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> 7, so rad(504) = 2 <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> 3 <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> 7 = 42.</p>
<p>We shall define the triplet of positive integers (<i>a</i>, <i>b</i>, <i>c</i>) to be an abc-hit if:</p>
<ol>
<li>GCD(<i>a,</i> <i>b</i>) = GCD(<i>a</i>, <i>c</i>) = GCD(<i>b</i>, <i>c</i>) = 1</li>
<li><i>a</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <i>b</i></li>
<li><i>a</i> + <i>b</i> = <i>c</i></li>
<li>rad(<i>abc</i>) <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <i>c</i></li>
</ol>
<p>For example, (5, 27, 32) is an abc-hit, because:</p>
<ol>
<li>GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1</li>
<li>5 <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 27</li>
<li>5 + 27 = 32</li>
<li>rad(4320) = 30 <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 32</li>
</ol>
<p>It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for <i>c</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 1000, with <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /><i>c</i> = 12523.</p>
<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='&sum;' border='0' style='vertical-align:middle;' /><i>c</i> for <i>c</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 120000.</p>
<p class="info">
Note: This problem has been changed recently, please check that you are using the right parameters.
</p>
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